On partial Galois Azumaya extensions
Let α be a globalizable partial action of a finite group G over a unital ring R, A=R⋆αG the corresponding partial skew group ring, Rα the subring of the α-invariant elements of R and α⋆ the partial inner action of G (induced by α) on the centralizer CA(R) of R in A. In this paper we present equivale...
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irk-123456789-1548402019-06-17T01:31:14Z On partial Galois Azumaya extensions Paques, A. Freitas, D. Let α be a globalizable partial action of a finite group G over a unital ring R, A=R⋆αG the corresponding partial skew group ring, Rα the subring of the α-invariant elements of R and α⋆ the partial inner action of G (induced by α) on the centralizer CA(R) of R in A. In this paper we present equivalent conditions to characterize R as an α-partial Galois Azumaya extension of Rα and CA(R) as an α⋆-partial Galois extension of the center C(A) of A. In particular, we extend to the setting of partial group actions similar results due to R. Alfaro and G. Szeto [1,2,3]. 2011 Article On partial Galois Azumaya extensions / D. Freitas, A. Paques // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 2. — С. 64–77. — Бібліогр.: 19 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16H05, 16S35, 16W22. http://dspace.nbuv.gov.ua/handle/123456789/154840 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let α be a globalizable partial action of a finite group G over a unital ring R, A=R⋆αG the corresponding partial skew group ring, Rα the subring of the α-invariant elements of R and α⋆ the partial inner action of G (induced by α) on the centralizer CA(R) of R in A. In this paper we present equivalent conditions to characterize R as an α-partial Galois Azumaya extension of Rα and CA(R) as an α⋆-partial Galois extension of the center C(A) of A. In particular, we extend to the setting of partial group actions similar results due to R. Alfaro and G. Szeto [1,2,3]. |
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Paques, A. Freitas, D. |
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Paques, A. Freitas, D. On partial Galois Azumaya extensions Algebra and Discrete Mathematics |
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Paques, A. Freitas, D. |
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Paques, A. |
| title |
On partial Galois Azumaya extensions |
| title_short |
On partial Galois Azumaya extensions |
| title_full |
On partial Galois Azumaya extensions |
| title_fullStr |
On partial Galois Azumaya extensions |
| title_full_unstemmed |
On partial Galois Azumaya extensions |
| title_sort |
on partial galois azumaya extensions |
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Інститут прикладної математики і механіки НАН України |
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2011 |
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On partial Galois Azumaya extensions / D. Freitas, A. Paques // Algebra and Discrete Mathematics. — 2011. — Vol. 11, № 2. — С. 64–77. — Бібліогр.: 19 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT paquesa onpartialgaloisazumayaextensions AT freitasd onpartialgaloisazumayaextensions |
| first_indexed |
2025-07-14T06:55:00Z |
| last_indexed |
2025-07-14T06:55:00Z |
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1837604379667988480 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 11 (2011). Number 2. pp. 64 – 77
c© Journal “Algebra and Discrete Mathematics”
On partial Galois Azumaya extensions
Daiane Freitas and Antonio Paques
Communicated by V. V. Kirichenko
Abstract. Let α be a globalizable partial action of a finite
group G over a unital ring R, A = R⋆αG the corresponding partial
skew group ring,Rα the subring of the α-invariant elements of R and
α⋆ the partial inner action of G (induced by α) on the centralizer
CA(R) of R in A. In this paper we present equivalent conditions
to characterize R as an α-partial Galois Azumaya extension of Rα
and CA(R) as an α⋆-partial Galois extension of the center C(A)
of A. In particular, we extend to the setting of partial group actions
similar results due to R. Alfaro and G. Szeto [1, 2, 3].
1. Introduction
The notion of Galois Azumaya extension was introduced by Alfaro and
Szeto in [3], motivated by an early work by themselves [2] about the
conditions necessary and sufficient for a skew group ring to be Azumaya.
Early, several other authors had considered this problem, among them
DeMeyer and Janusz [6] for group rings, Szeto and Wong [19] for twisted
group rings, and Ikehata [12] for skew group ring over commutative rings.
In [2] Alfaro and Szeto extend the Ikehata’s results to the noncommutative
case. The results in [2] were considered later by Ouyang [14] for smash
products, by Carvalho [5] for not twisted crossed products and by Paques
and Sant’Ana [16] for partial crossed products. All these above mentioned
results show in particular, each one in its respective context, that there
exists an interesting and closed relation among the notions of Azumaya
algebra, Galois extension and Hirata separability.
2000 Mathematics Subject Classification: 16H05, 16S35, 16W22.
Key words and phrases: partial group action, partial skew group ring, partial
Galois extension, partial Galois Azumaya extension.
D. Freitas, A. Paques 65
In this paper we deal with globalizable partial actions of a finite group
G over a ring R, and the respective partial skew group ring. Our aim is to
extend to this context the Alfaro-Szeto’s results from [1, 2]. First we prove
a result analogous to [2, Theorems 1 and 2] for the partial skew group ring
A = R ⋆α G, where α denotes a partial action of G on R (see Theorem
1.1). In the sequel we consider the “inner" partial action α∗, induced by
α, of G on the centralizer CA(R) of R in A (see Proposition 2.8) and we
also prove a result analogous to [1, Theorem 3] for the partial skew group
ring Λ = CA(R) ⋆α⋆ G (see Theorem 1.2).
Throughout, unless otherwise stated, rings and algebras are associative
and unital. For any non-empty subset X of a ring R, any subring Y of R
containing X and any (Y, Y )-bimodule V we will denote by CV (X) the
centralizer of X in V , that is, the set of all v ∈ V such that xv = vx for
all x ∈ X. If in particular X = Y = V = R, then CV (X) is the center of
R and we will denote it simply by C(R).
A partial action α of a group G on a ring R [7] is a pair
α = ({Dg}g∈G, {αg}g∈G),
where for each g ∈ G, Dg is an ideal of R and αg : Dg−1 → Dg is
an isomorphism of (nonnecessarily unital) rings, satisfying the following
conditions:
(i) D1 = R and α1 is the identity automorphism IR of R;
(ii) αg(Dg−1 ∩Dh) = Dg ∩Dgh;
(iii) αg ◦ αh(r) = αgh(r), for every r ∈ Dh−1 ∩D(gh)−1 .
If Dg = R for every g ∈ G, then α is a global action of the group G on R,
by automorphisms of R.
We will assume hereafter that every ideal Dg is unital, with its identity
element denoted by 1g (in particular, each 1g is a central idempotent
of R). By [7, Theorem 4.5], this condition is equivalent to say that α
has a globalization (or an enveloping action), which means that there
exist a (nonnecessarily unital) ring T and a global action of G on T , by
automorphisms βg (g ∈ G), such that R can be considered an ideal of T
and the following conditions hold:
(i) T =
∑
g∈G βg(R);
(ii) Dg = R ∩ βg(R), for all g ∈ G;
(iii) αg = βg|D
g−1 .
66 On partial Galois Azumaya extensions
In particular, under these conditions, we have R = T1R and
1g = 1Rβg(1R), αg(r1g−1) = βg(r)1R and αg(1h1g−1) = 1g1gh
for every g, h ∈ G and r ∈ R. Also, if G is finite then T is unital.
Following [7], the partial skew group ring R⋆αG is defined as the direct
sum ⊕
g∈G
Dgδg,
where the δ′gs are symbols, with the usual sum and the multiplication
defined by the rule
(rδg)(sδh) = rαg(s1g−1)δgh
for all g, h ∈ G, r ∈ Dg and s ∈ Dh. Since everyDg is unital by assumption,
then R ⋆α G is associative (see [7, Proposition 2.5 and Theorem 3.1]) and
unital, with the identity element given by 1Rδ1. Also, R ⋆α G is a ring
extension of R via the embedding r 7→ rδ1, for all r ∈ R.
The subring of invariants of R under α [8] is defined as
Rα = {r ∈ R : αg(r1g−1) = r1g, for all g ∈ G},
Note that if α is global then Rα = RG as usual. We say that R is an
α-partial Galois extension of Rα (or a G-Galois extension of RG, if α is
global) if G is finite and there exists a finite set {xi, yi}
m
i=1 of elements of
R such that
∑m
i=1 xiαg(yi1g−1) = δ1,g1R, for every g ∈ G. Such a set is
called a partial Galois coordinate system of R over Rα.
A non-empty subset X of R is called α-invariant (or G-invariant, if
α is global) if αg(Dg−1 ∩ X) = Dg ∩ X, for every g ∈ G. In particular,
the centralizer CR(X) of any non-empty α-invariant subset X of R is also
α-invariant. Since every Dg = R1g it is immediate to see that if X is an
α-invariant subring of R then 1g belongs to X for every g ∈ G. Moreover,
in this case α induces by restriction a partial action on X ′ = CR(X) given
by the pair ({X ′1g}g∈G, {αg|X′1
g−1}g∈G).
Let S ⊇ R be a ring extension. We say that S is Hirata-separable over
R [9] if S ⊗R S is isomorphic, as an S-bimodule, to a direct summand of
a finite direct sum of copies of S or, equivalently, if there exist elements
xi ∈ CS(R) and yi ∈ CS⊗RS(S), 1 ≤ i ≤ m, such that
∑
1≤i≤m xiyi =
1S ⊗ 1S [18, Proposition 1]. The set {xi, yi | 1 ≤ i ≤ m} is called an
Hirata-separable system of S over R. Hirata-separable extensions are
separable [10, Theorem 2.2]. S is called separable over R (see [11]) if
the multiplication map mS : S ⊗R S → S is a splitting epimorphism of
S-bimodules or, equivalently, if there exists an element x ∈ CS⊗RS(S)
D. Freitas, A. Paques 67
such that mS(x) = 1S . If R ⊆ C(S) (resp., R = C(S)) we also say that S
is a separable (resp., an Azumaya) R-algebra. A ring R is called Azumaya
if it is an Azumaya C(R)-algebra. Furthermore, provided the existence
of a partial action α of a finite group G on a ring R, we say that R is
an α-partial Galois Azumaya extension of Rα if R is an α-partial Galois
extension of Rα, Rα is an Azumaya ring and C(Rα) = C(R)α.
Our main results in this paper are the following theorems.
Theorem 1.1. Let α be a globalizable partial action of a finite group G on
a ring R, and A = R ⋆α G. Then the following statements are equivalent:
(i) R is an α-partial Galois Azumaya extension of Rα.
(ii) A is Azumaya and C(A) ⊆ R.
(iii) A is Hirata-separable over R, R is a separable C(A)-algebra, and
C(A) = C(R)α.
(iv) CR(R
α) is an α-partial Galois extension of C(A) and Rα is an
Azumaya C(A)-algebra.
Moreover, in this case, C(A) = C(R)α = C(Rα) and A ≃ Rα ⊗C(A)
EndC(A)(CR(R
α)).
Theorem 1.2. Let α be a globalizable partial action of a finite group G on
a ring R, and A = R ⋆α G. Let α⋆ be the partial inner action, induced by
α, of G on CA(R), and Λ = CA(R) ⋆α⋆ G. Then the following statements
are equivalent:
(i) CA(R) is an α⋆-partial Galois extension of C(A).
(ii) Λ is Hirata-separable over CA(R).
(iii) A is Hirata-separable over R and CΛ(CΛ(CA(R))) = CA(R).
(iv) Λ is an Azumaya C(A)-algebra.
Moreover, in this case,C(Λ) = C(CA(R))
α⋆
= C(R)α = C(A) = CA(R)
α⋆
.
Their proofs will be done via an explicit way going from the partial to
the global case and conversely (see section 3). Some examples illustrating
these results are given in the section 4.
We present in the next section the necessary preparation to prove the
above theorems. Actually, the proofs of these theorems can be seen as
applications of the results we will present in the next section, which also
are of some independent interest.
68 On partial Galois Azumaya extensions
2. Prerequisites
From now on, G will denote a finite group and α = ({Dg}g∈G, {αg}g∈G)
a partial action of G on a given ring R, with globalization (T, β). Also,
let A = R ⋆α G and B = T ⋆β G.
Since T =
∑
g∈G βg(R), putting G = {g1 = 1, g2, . . . , gn}, we have
that 1T = e1 ⊕ e2 ⊕ · · · ⊕ en, where e1 = 1R and ei = (1T − 1R) · · · (1T −
βgi−1(1R))βgi(1R), for every 2 ≤ i ≤ n, (see [8]). Let ψ : T → T be the
map given by
ψ(x) =
n∑
i=1
βgi(x)ei =
∑
1≤l≤n
∑
i1<···<il
(−1)l+1βgi1 (1R) · · ·βgil−1
(1R)βgl(x),
for every x ∈ T . Such a map was introduced in [8], it is clearly (left and
right) TG-linear and multiplicative, and it will be useful in the sequel.
Lemma 2.1. The following statements are equivalent:
(i) R is an α-partial Galois Azumaya extension of Rα.
(ii) T is a G-Galois Azumaya extension of TG.
Proof. See [15, Corollary 2.3].
Lemma 2.2. The following statements are equivalent:
(ii) A is Azumaya and C(A) ⊆ R.
(ii) A is Hirata-separable over R, R is a separable C(R)α-algebra and
C(A) = C(R)α.
Proof. We start by observing that C(A) ⊆ R if and only if C(A) =
C(R)α. Indeed, if r ∈ C(R)α then (rgδg)(rδ1) = rgαg(r1g−1)δg = rgrδg =
(rδ1)(rgδg), for every g ∈ G and rg ∈ Dg. So C(R)α ⊆ C(S). Conversely, if
x ∈ C(S) ⊆ R then x ∈ C(R) and x(1gδg) = (xδ1)(1gδg) = (1gδg)(xδ1) =
αg(x1g−1)δg, which implies α(x1g−1) = x1g, for every g ∈ G. So x ∈
C(R)α.
Now, the proof of the equivalence (i)⇔(ii) above follows by tracking
the same arguments as in the global case. For the details, see the proof
of (1)⇒(2) and the first paragraph of the proof of (2)⇒(3) of Theorem 1
in [2].
Lemma 2.3. The following statements are equivalent:
(i) A is Hirata-separable over R.
D. Freitas, A. Paques 69
(ii) B is Hirata-separable over T .
Proof. (i)⇒ (ii) We start by observing that C(R) ⊆ C(T ) and ψ(C(R)) ⊆
C(T ). The first assertion is immediate since for any r ∈ C(R) and any
t ∈ T we have rt = r(1Rt) = (1Rt)r = tr. For the second assertion, taking
r ∈ C(R) and putting x = ψ(r) ∈ T , it is clear that ψ(s)x = xψ(s)
for every s ∈ R and thus
∑
i βgi(s)eix =
∑
i xβgi(s)ei, which implies
βgi(s)eix = xβgi(s)ei, for all 1 ≤ i ≤ n. Since T =
⊕
i βgi(R)ei the result
follows.
We also have that CA(R) ⊆ CB(T ). Indeed, recall from [16, Lemma
2.1] that CA(R) =
∑
g∈G φg(R)δg, where φg(R) is the set of all r ∈ Dg
such that rαg(s1g−1) = sr, for all s ∈ R. So, it is enough to show that
φg(R) ⊆ φg(T ) for all g ∈ G, which is immediate since given r ∈ φg(R)
we have rβg(t) = r1gβg(t) = r1Rβg(1Rt) = rαg(1Rt1g−1) = 1Rtr = tr, for
all t ∈ T .
Now suppose that A is Hirata-separable over R and take xi ∈ CA(R)
and yi ∈ CA⊗RA(A), 1 ≤ i ≤ m, such that
∑
i xiyi = 1A ⊗ 1A. Thus,
each xi =
∑
g∈G ri,gδg, with ri,g ∈ φg(R) for all g ∈ G. In particular
ri,1 ∈ C(R). Also, by [16, Lemma 2.1(ii)-(iv)] we can take each yi =∑
h∈G αh(ci1h−1)δh ⊗ 1h−1δh−1 , with ci ∈ C(R). From
∑
i xiyi = 1A ⊗ 1A
we get
∑
i ri,1ci = 1R and
∑
i ri,gαgh(ci1(gh)−1) = 0 if either g 6= 1 or
h 6= 1.
Then, denoting ri,l := ri,gl we have
∑
i ri,lβglgj (ci)βgl(1T − ej) =
∑
i ri,l1Rβglgj (ci)βgl(1T − ej)
= (
∑
i ri,lαglgj (ci1(glgj)−1))βgl(1T − ej)
= 0
for all 2 ≤ j, l ≤ n,
∑
i ψ(ri,1)βgj (ci)(1T − ej) =
∑
i,k βgk(ri,1)ekβgj (ci)(1T − ej)
=
∑
k βgk(
∑
i ri,1βg−1
k
gj
(ci))ek(1T − ej)
=
∑
k βgk(
∑
i ri,1αg−1
k
gj
(ci1g−1
j gk
))ek(1T − ej)
= βgj (1R)ej(1T − ej)
= 0
for all j 6= 1, and
∑
i ri,lβgl(ψ(ci)) =
∑
i ri,l1glβgl(ψ(ci))
=
∑
i ri,l1Rβgl(ψ(ci)1R)
=
∑
i ri,l1Rβgl(ci)
=
∑
i ri,lαgl(ci1g−1
l
)
= 0
70 On partial Galois Azumaya extensions
for all l 6= 1.
Taking ui = ψ(ri,1)δ1 +
∑
2≤l≤n ri,lδgl and vi = ψ(ci)δ1 ⊗ 1T δ1 +∑
2≤j≤n βgj (ci)(1T − ej)δgj ⊗ 1T δg−1
j
, we have from the above that ui ∈
CB(T ), vi ∈ CB⊗TB(B) for each 1 ≤ i ≤ m, and
∑
i uivi = ψ(
∑
i ri,1ci)δ1⊗
1T δ1 = ψ(1R)δ1 ⊗ 1T δ1 = 1T δ1 ⊗ 1T δ1 = 1B ⊗ 1B. Hence, B is Hirata-
separable over T .
(ii)⇒ (i) Let ui ∈ CB(T ) and vi ∈ CB⊗TB(B), 1 ≤ i ≤ m, be a Hirata-
separable system of B over T . Then by [16, Lemma 2.1] ui =
∑
g∈G ti,gδg
with ti,g ∈ φg(T ), and we can take vi =
∑
g∈G βg(di)δg ⊗ 1T δg−1 with
di ∈ C(T ), for every 1 ≤ i ≤ m. It is immediate that ci = di1R ∈ C(R)
for each 1 ≤ i ≤ m. Also, every ri,g = ti,g1g ∈ φg(R) since ri,g ∈ Dg
and ri,gαg(r1g−1) = ri,gβg(r) = 1gti,gβg(r) = 1grti,g = rri,g for all r ∈ R.
Again by [16, Lemma 2.1] we have xi =
∑
g∈G ri,gδg ∈ CA(R) and yi =∑
g∈G αg(ci1g−1)δg ⊗ 1g−1δg−1 ∈ CA⊗RA(A).
Notice that the bi-additive map B ×B → A⊗R A, given by (b, b′) 7→
1Ab1A ⊗ 1Ab
′1A is T -balanced, and so induces a well-defined left T -linear
map θ : B ⊗T B → A⊗R A.
Therefore,
∑
i xiyi =
∑
i
∑
g,h∈G(ri,gδg)(αh(ci1h−1)δh)⊗ 1h−1δh−1
=
∑
i
∑
g,h∈G ri,gαg(αh(ci1h−1)1g−1)δgh ⊗ 1h−1δh−1
=
∑
i
∑
g,h∈G ri,gαgh(ci1(gh)−1)δgh ⊗ 1h−1δh−1
=
∑
i
∑
g,h∈G ti,g1gαgh((di1R)1(gh)−1)δgh ⊗ 1h−1δh−1
=
∑
i
∑
g,h∈G ti,gαgh((di1R)1(gh)−1)δgh ⊗ 1h−1δh−1
=
∑
i
∑
g,h∈G ti,g1Rβgh(di1R)δgh ⊗ 1h−1δh−1
=
∑
i
∑
g,h∈G 1A(ti,gβgh(di)δgh)1A ⊗ 1A(1T δh−1)1A
= θ(
∑
i
∑
g,h∈G ti,gβgh(di)δgh ⊗ 1T δh−1)
= θ(1B ⊗ 1B)
= 1A1B1A ⊗ 1A1B1A
= 1A ⊗ 1A
and A is Hirata-separable over R.
Lemma 2.4. The following statements are equivalent:
(i) R is a separable C(R)α-algebra.
(ii) T is a separable C(T )G-algebra.
Proof. It follows from [15, Lemma 2.1 (i), (iii) and (ix)].
Corollary 2.5. The following statements are equivalent:
(i) A is Azumaya and C(A) ⊆ R.
D. Freitas, A. Paques 71
(ii) B is Azumaya and C(B) ⊆ T .
Proof. It follows from Lemmas 2.2, 2.3 and 2.4, and [2, Theorem 1].
Lemma 2.6. The following statements are equivalent:
(i) Rα is Azumaya.
(ii) TG is Azumaya.
Proof. It follows from [15, Lemma 2.1 (iii) and (viii)].
Lemma 2.7. The following statements are equivalent:
(i) CR(R
α) is an α-partial Galois extension of C(Rα).
(ii) CT (T
G) is a G-Galois extension of C(TG).
Proof. It follows from [15, Lemma 2.1(ii)-(iii) and Lemma 2.4] that
(CT (T
G), β′), with β′g = βg|
CT (TG)
for all g ∈ G, is a globalization of
(CR(R
α), α′), with α′
g = αg|CR(Rα)
for all g ∈ G. Now the result follows
from [8, Theorem 3.3].
These above listed results are sufficient to prove Theorem 1.1 (see
section 3). For the proof of Theorem 1.2 we also need to introduce the
notion of an “inner" partial action, denoted α⋆, of G on CA(R), induced
by the partial action α on R. In the particular case that CA(R) ⊆ R, the
partial action α⋆ coincides with the restriction of α to CA(R).
Proposition 2.8. Under the conditions above assumed, we have that:
(i) β induces an action β⋆ of G on B by inner automorphisms β⋆g , given
by β⋆g(tδh) = (1T δg)(tδh)(1T δg−1) = βg(t)δghg−1, for all g, h ∈ G
and t ∈ T ,
(ii) β⋆g (1g−1) = 1g, β⋆g (1g−11h) = 1g1gh, and 1Rβ
⋆
g (r) = αg(r1g−1), for
g, h ∈ G and r ∈ R,
(iii) CB(T ) is β⋆-invariant, and therefore β⋆ induces, by restriction, an
action of G on CB(T ),
(iv) CA(R) = CB(T )1R = CB(T )1A,
(v) α⋆ := ({CA(R)g = CA(R)1g}g∈G, {α⋆
g = β⋆g |CA(R)
g−1
}g∈G) is a
partial action of G on CA(R), and α⋆
g(a) = (1gδg)a(1g−1δg−1), for
all g ∈ G and a ∈ CA(R)g−1 ,
72 On partial Galois Azumaya extensions
(vi) (CB(T ), β
⋆|CB(T )) is a globalization for (CA(R), α
⋆),
(vii) CA(R)
α⋆
:= {a ∈ CA(R) | α
⋆
g(a1g−1) = a1g, for all g ∈ G} = C(A).
Proof. (i) It is enough to see that the map β⋆ : G→ Aut(B), g 7→ β⋆g , is a
homomorphism of groups. Clearly, β⋆ is well-defined, and for all g, h, l ∈ G
and t ∈ T we have
β⋆(gh)(tδl) = β⋆gh(tδl) = βgh(t)δghl(gh)−1 = βg(βh(t))δg(hlh−1)g−1
= β⋆g (βh(t)δhlh−1) = β⋆g (β
⋆
h(tδl)) = β⋆(g) ◦ β⋆(h)(tδl)
(ii) For every r ∈ R and g, h ∈ G we have
β⋆g (1g−1) = β⋆g (1g−1δ1) = βg(1Rβg−1(1R))δ1
= βg(1R)1Rδ1 = 1gδ1 = 1g.
β⋆g (1g−11h) = β⋆g (1Rβg−1(1R)βh(1R)δ1) = βg(1Rβg−1(1R)βh(1R))δ1
= βg(1R)1Rβgh(1R)δ1) = 1g1ghδ1 = 1g1gh.
1Rβ
⋆
g (r) = 1Rβ
⋆
g (rδ1) = 1Rβg(r)δ1 = αg(r1g−1)δ1 = αg(r1g−1).
(iii) For every b ∈ CB(T ), g ∈ G and t ∈ T we have
β⋆g (b)t = (1T δg)b(1T δg−1)(tδ1) = (1T δg)b(βg−1(t)δg−1)
= (1T δg)b(βg−1(t)δ1)(1T δg−1) = (1T δg)(βg−1(t)δ1)b(1T δg−1)
= βg(βg−1(t))(1T δg)b(1T δg−1) = tβ⋆g (b),
so β⋆g (b) ∈ CB(T ).
(iv) First note that
1RB1R = 1R(
⊕
g∈G
Tδg)1R =
⊕
g∈G
(T1Rδg)(1Rδ1)
=
⊕
g∈G
T1Rβg(1R)δg =
⊕
g∈G
T1gδg =
⊕
g∈G
Dgδg = A.
Also, 1R is clearly a central idempotent in CB(T ). Hence, CB(T )1R =
1RCB(T )1R = C1RB1R(T ) = CA(T ) ⊆ CA(R). For the reverse inclusion,
observe that given a ∈ CA(R) there exists b ∈ B such that a = 1Rb1R
and so at = (a1R)t = a(1Rt) = (1Rt)a = t(1Ra) = ta, for every t ∈ T .
(v) Clearly, each CA(R)g is an ideal of CA(R) and
α⋆
g(CA(R)g−1) = β⋆g (CA(R)1g−1) = β⋆g (CB(T )1g−1)
D. Freitas, A. Paques 73
= β⋆g (CB(T ))β
⋆
g (1g−1) = CB(T )1g
= CA(R)1g = CA(R)g.
Thus, each α⋆
g : CA(R)g−1 → CA(R)g is an isomorphism of rings. It is
straightforward to check that the three conditions for α⋆ to be a partial
action are also satisfied. And
(1gδg)a(1g−1δg−1) = (1Tαg(1g−1)δg)a(1g−11T δg−1)
= (1T δg)(1g−1a1g−1)(1T δg−1) = β⋆g (a) = α⋆
g(a),
for all g ∈ G and a ∈ CA(R)g−1 .
(vi) Let G = {g1 = 1, g2, . . . , gn}. It follows from (iv) that β⋆gi(CA(R))
is an ideal of CB(T ) for all 1 ≤ i ≤ n. Since 1A = 1Rδ1 ∈ CA(R), we have
1B = 1T δ1 =
∑
i βgi(1R)eiδ1 =
∑
i(βgi(1R)δ1)(eiδ1) =
∑
i β
⋆
gi
(1A)eiδ1 ∈∑
i β
⋆
gi
(CA(R)), and consequently CB(T ) =
∑
i β
⋆
gi
(CA(R)). Finally,
CA(R) ∩ β
⋆
g (CA(R)) = CB(T )1R ∩ β⋆g (CB(T )1R)
= CB(T )1R ∩ β⋆g (CB(T ))β
⋆
g (1R)
= CB(T )1R ∩ CB(T )β
⋆
g (1R) = CB(T )1Rβ
⋆
g (1R)
= CB(T )1g = CA(R)1g = CA(R)g.
(vii) It is immediate, from the definitions of centralizer and subring
of α⋆-invariants, that x ∈ CA(R)
α⋆
if and only if (rδg)x = x(rδg) for all
g ∈ G and r ∈ Dg if and only if x ∈ C(A).
Lemma 2.9. Let Γ = CB(T ) ⋆β⋆ G, Λ = CA(R) ⋆α⋆ G, C0(Γ) = CB(T )
(resp., C0(Λ) = CA(R)) and Ci(Γ) = CΓ(Ci−1(Γ)) (resp., Ci(Λ) =
CΛ(Ci−1(Λ))) for all i ≥ 1. Then,
(i) 1RΓ1R = Λ,
(ii) 1R is a central idempotent in Ci(Γ) and
(iii) Ci(Γ)1R = Ci(Λ) for all i ≥ 0.
Proof. It follows by induction via the same arguments used in the proof
of Proposition 2.8(iv).
Remark 2.10. Note that α⋆ induces an inner partial action, denoted α⋆⋆,
of G on CΛ(CA(R)) in the same fashion that α induces α⋆. Therefore,
Proposition 2.8 also applies in this similar situation. Furthermore, the
restriction of α⋆⋆ to CA(R) (resp. R) coincides with α⋆ (resp. α).
74 On partial Galois Azumaya extensions
3. The Proofs
Proof of Theorem 1.1:
(i)⇒(ii) It follows from Lemma 2.1, [3, Theorem 1, (3)⇒(1)], and Corol-
lary 2.5.
(ii)⇒(iii) It follows from Lemma 2.2.
(iii)⇒(i) If follows from Lemmas 2.3 and 2.4, [2, Theorem 1, (2)⇒(3)],
and Lemma 2.1.
(i)⇒(iv) It follows from Lemma 2.1, [2, Theorem 2(3)], and Lemmas 2.6
and 2.7.
(iv)⇒(i) It is enough to notice that any partial Galois coordinate
system of CR(R
α) over C(A) is also a partial Galois coordinate system of
R over Rα.
For the last assertion we observe that by the same arguments used in the
proof of [2, Theorem 2(1)] we have R⋆αG ≃ Rα⊗C(A)(CR(R
α)⋆αG). Since
CR(R
α) is an α-partial Galois extension of C(A), then CR(R
α) ⋆α G ≃
EndC(A)(CR(R
α)) by [8, Theorem 3.3], and the result follows.
Proof of Theorem 1.2:
(i)⇔(ii) By Proposition 2.8(vi) (CB(T ), β
⋆) is a globalization of (CA(R), α
⋆)
and by [8, Theorem 3.3], CB(T ) is a Galois extension of CB(T )
β⋆
(= C(B))
if and only if CA(R) is an α⋆-partial Galois extension of CA(R)
α⋆
(= C(A)).
Then, the result follows from [8, Theorem 3.3], [1, Theorem 2], and
Lemma 2.3.
(ii)⇒(iii) It follows from Lemma 2.3, [1, Theorem 3] and Lemma 2.9.
(iii)⇒(iv) It follows by some arguments similar to the corresponding
ones used in the proof of [1, Proposition 3], as we will see. From [10, Theo-
rem 2.2] we have that A is a separable extension of R and so, by [4, Propo-
sition 3.1] there exists c ∈ C(R) such that tα(c) :=
∑
g∈G αg(c1g−1) = 1R.
We also have from [17, Proposition 1.2] that CA(CA(R)) = R, for R is a
direct summand of A as a left R-module. Hence,
C(A) = C(R)α
(since C(A) ⊆ CA(CA(R)) (see the proof of Lemma 2.2)),
C(R) = R ∩ CA(R) = CA(CA(R)) ∩ CA(R) = C(CA(R))
and
trα⋆(cδ1) :=
∑
g∈G
α⋆
g((cδ1)1g−1) =
∑
g∈G
β⋆g (c1g−1δ1)
D. Freitas, A. Paques 75
=
∑
g∈G
βg(c1g−1)δ1 =
∑
g∈G
αg(c1g−1)δ1
= trα(c)δ1 = 1Rδ1 = 1A,
which implies, again by [4, Proposition 3.1], that Λ is separable over
CA(R). Since CA(R) is separable over C(A) by [13, Lemma 1], then Λ is
also separable over C(A) by the transitivity of the separability. Thus, it
remains to prove that C(Λ) = C(A). Indeed, setting L = CΛ(CA(R)), it
follows by Proposition 2.8 that C(Λ) = Lα⋆⋆
. Furthermore, it is immediate
to see that Lα⋆⋆
= C(L)α
⋆⋆
and then
C(Λ) = C(L)α
⋆⋆
= (CΛ(L) ∩ L)
α⋆⋆
= (CΛ(CΛ(CA(R))) ∩ L)
α⋆⋆
= (CA(R) ∩ L)
α⋆
= (CA(R) ∩ CΛ(CA(R)))
α⋆
= C(CA(R))
α⋆
= C(R)α = C(A)
(iv)⇒ (i) It follows from Corollary 2.5, [1, Proposition 3] and [8,
Theorem 3.3].
The last assertion is immediate from the above.
Remark 3.1. The equivalences (i)⇔(ii)⇔(iv) of Theorem 1.2 can also
be seen as a corollary of Theorem 1.1 since CA(R), under the condition
(i), is an α⋆-partial Galois Azumaya extension of CA(R)
α⋆
indeed.
4. Some Examples
Example 4.1. Let R be an Azumaya ring, G a finite group and α =
({Dg}g∈G, {αg}g∈G) the trivial partial action of G on R, that is, D1 = R,
α1 = IR and Dg = 0 = αg for all g 6= 1. In this case we have that
A = R ⋆α G = R = Rα, C(A) = C(R) = C(R)α = C(Rα) and R is an
α-partial Galois Azumaya extension of Rα. Furthermore, CA(R) = C(R),
α⋆ coincides with the restriction of α to C(R), Λ = CA(R) ⋆α⋆ G = C(R),
and each one of the equivalent assertions of Theorems 1.1 and 1.2 is
trivially satisfied.
Example 4.2. Let S be an Azumaya ring and set T =
∏
1≤i≤4
Si =
⊕
1≤i≤4
Siei, where Si = S for every 1 ≤ i ≤ 4, and each ei is the quadruple
whose jth-coordinate is 1S if j = i and zero otherwise.
Consider G a cyclic group of order 4 generated by g, and β : G →
Aut(T ) a global action of G on T by automorphisms βgi given by∑
1≤j≤4 ajej 7→
∑
1≤j≤4 ajej+i(mod 4). Take R = Se1 ⊕ Se3 ⊕ Se4 and
α the partial action of G on R obtained from β by restriction to R, that
is, α = ({Dgi}1≤i≤4, {αgi}1≤i≤4) where
76 On partial Galois Azumaya extensions
D1 = Dg0 = R, Dg = Se1, Dg2 = Se1 ⊕ Se3, Dg3 = Se4,
α1 = αg0 = IR,
αg : Dg3 → Dg, se4 7→ se1,
αg2 : Dg2 → Dg2 , se1 + s′e3 7→ s′e1 + se3,
αg3 : Dg → Dg3 , se1 7→ se4
Clearly (T, β) is a globalization of (R,α) and it is straightforward to
verify that Rα = S(e1 + e3 + e4) ≃ S and C(Rα) = C(R)α = C(S)(e1 +
e3 + e4) ≃ C(S). It is also clear that R is an α-partial Galois extension
of Rα. In this case a partial Galois coordinate system is given by x1 =
y1 = e1, x2 = y2 = e3 and x3 = y3 = e4. Hence, R is an α-partial
Galois Azumaya extension of Rα. Consequently, A = R ⋆α G is Azumaya
with C(A) = C(R)α by Theorem 1.1. Moreover, it is easy to check that
CA(R) = C(R) and so α⋆ is the restriction of α to C(R). Since each
ei ∈ C(R), i = 1, 3, 4, then CA(R) is also an α-partial Galois Azumaya
extension of C(R)α.
Example 4.3. (see [8, Example 6.3]) Let S be a commutative ring and
G is a cyclic group of order 6, generated by g. Assume that S is a (global)
Galois extension of SG and set R =
∏
1≤i≤5
Si =
⊕
1≤i≤5
Siei, where Si = S
for every 1 ≤ i ≤ 5, and each ei is the quintuple whose jth-coordinate is
1S if j = i and zero otherwise. Taking Dg0 = R, Dgi = Se6−i = Re6−i,
αg0 = IR and αgi(sei) = gi(s)e6−i, 1 ≤ i ≤ 5, it is straightforward to
check that α = ({Dgi}
5
i=0, {αgi}
5
i=0) is a partial action of G on R and
Rα = {s1e1 + s2e2 + s3e3 + σ2(s2)e4 + σ(s1)e5 | s1, s2 ∈ S, s3 ∈ Sσ3
}. Let
si, ti ∈ S, 1 ≤ i ≤ m, be a Galois coordinate system for S over SG and
consider the elements xj = yj = ej , j = 1, 2, 4, 5 together with the elements
xi3 = sie3, yi3 = tie3. It is easy to see that this gives a partial Galois
coordinate system forR overRα. Since, in addition,R is commutative, then
R is an α-partial Galois Azumaya extension of Rα. By Theorem 1.1, A =
R⋆αG is Azumaya and C(A) = Rα. Furthermore, CA(R) = Rδg0 ⊕Se3δg3
and it is an α⋆-partial Galois extension of CA(R)
α⋆
= C(A) with the
partial Galois coordinate system given by the elements uj = vj = ejδg0 ,
j = 1, 2, 4, 5 and ui3 = sie3δg0 , vi3 = tie3δg0 , 1 ≤ i ≤ m.
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Contact information
D. Freitas Instituto de Matemática, Estat́istica e F́isica,
Universidade Federal do Rio Grande, 96201-900,
Rio Grande, RS, Brazil
E-Mail: daianefreitas@furg.br
A. Paques Instituto de Matemática, Universidade Federal
do Rio Grande do Sul, 91509-900, Porto Alegre,
RS, Brazil
E-Mail: paques@mat.ufrgs.br
Received by the editors: 25.04.2011
and in final form 06.05.2011.
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